Hartree-Fock self-consistent field Slater determinant

There are two ways to improve the accuracy in order to obtain solutions to almost any degree of accuracy. The first is via the so-called self-consistent field-Hartree-Fock (SCF-HF) method, which is a method based on the variational principle that gives the optimal one-electron wave functions of the Slater determinant. Electron correlation is, however, still neglected (due to the assumed product of one-electron wave functions). In order to obtain highly accurate results, this approximation must also be eliminated.6 This is done via the so-called configuration interaction (Cl) method. The Cl method is again a variational calculation that involves several Slater determinants. 

The Hartree-Fock or self-consistent field (SCF) method is a procedure for optimizing the orbital functions in the Slater determinant (9.1), so as to minimize the energy (9.4). SCF computations have been carried out for all the atoms of the periodic table, with predictions of total energies and ionization energies generally accurate in the 1-2% range. Fig. 9.2 shows the electronic radial distribution function in the argon atom, obtained from a Hartree-Fock computation. The shell structure of the electron cloud is readily apparent. 

The most uniformly successful family of methods begins with the simplest possible n-electron wavefunction satisfying the Pauli antisymmetry principle - a Slater determinant [2] of one-electron functions % r.to) called spinorbitals. Each spinorbital is a product of a molecular orbital xpt(r) and a spinfunction a(to) or P(co). The V /.(r) are found by the self-consistent-field (SCF) procedure introduced [3] into quantum chemistry by Hartree. The Hartree-Fock (HF) [4] and Kohn-Sham density functional (KS) [5,6] theories are both of this type, as are their many simplified variants [7-16],... 

The self-consistent field Hartree-Fock (HF) method is the foundation of AI quantum chemistry. In this simplest of approaches, the /-electron ground state function T fxj,. X/y) is approximated by a single Slater determinant built from antisymmetrized products of one-electron functions i/r (x) (molecular orbitals, MOs, X includes space, r, and spin, a, = 1/2 variables). MOs are orthonormal single electron wavefunctions commonly expressed as linear combinations of atom-centered basis functions ip as i/z (x) = c/ii /J(x). The MO expansion coefficients are... 

II Another relatively simple approximation is the Hartree-Fock (H-F) self-consistent field (SCF) method, which yields the best single Slater determinant for a basis function of the separable type given in Eq. (77). However, this best result is generally unsatisfactory for molecular problems, except possibly if it turns out that a correction from more exact computation for one or a few nuclear configurations can be smoothly applied to the approximate result. For a discussion of the H-F (SCF) method see Reference 85. 

The Multi-Configuration Self-Consistent Field method combines the ideas of orbital optimization through a SCF technique as in the Hartree-Fock method, and a multiconfiguration expansion of the electronic wavefunction as in the configuration interaction method. In other words, the electronic wavefunction is still expressed as a linear combination of Slater determinants but now both the coeffi-... 

The technique was applied to the atoms of He, Rb, Na, Cl . And it was the justification of Hartree s method that got Slater (1929) to think more about the theory of complex spectra, introducing determinants and the variational method for deriving analytically the self-consistent field equations with the right symmetry properties, as we have already discussed in chapter 2. Furthermore, Vladimir Fock (1930) also... 

All the ab initio methods discussed here are based on the Hartree-Fock (HF) or self-consistent field method. In closed-shell HF theory the unperturbed many-electron wavefunction is approximated by a single Slater determinant... 

As a matter of fact, as in the Hartree-Fock (HF) scheme, the KS equation is a pseudo-eigenvalue problem and has to be solved iteratively through a self-consistent field procedure to determine the charge density p(r) that corresponds to the lowest energy. The self-consistent solutions 4>ia resemble those of the HF equations. Still, one should keep in mind that these orbitals have no physical significance other than in allowing one to constitute the charge density. We want to stress that the DFT wavefunction is not a Slater determinant of spin orbitals. In fact, in a strict sense there is no A -electron wavefunction available in DFT. ... 

In other words, the potential term in the equation determining the singleparticle function ij/, is to be found self-consistently from the mean field of the other electrons. The Hartree-Fock (HF) approximation incorporates Fermi statistics into this picture by replacing the product wavefunction (Eq. (1)) by a single determinantal function (Slater determinant) constructed from the tjf,. In this case, the single-particle functions satisfy an equation similar to (2) but with an additional non-local potential term. 

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